We study the purchasing power parity (PPP) puzzle in a multisector, two-country, sticky-price model. Firms' price stickiness differs across sectors, in accordance with recent microeconomic evidence on price setting in various countries. Combined with local currency pricing, these differences lead sectoral real exchange rates to exhibit heterogeneous dynamics. We show that in this economy, deviations of the real exchange rate from PPP are more volatile and persistent when compared with a counterfactual one-sector world economy that features the same average frequency of price changes and is otherwise identical to the multisector world economy. When simulated with a sectoral distribution of price stickiness that matches the microeconomic evidence for the U.S. economy, the model produces a half-life of deviations from PPP of forty-five months. In contrast, the half-life of such deviations in the counterfactual one-sector economy is only slightly above one year. As a by-product, our model provides a decomposition of this difference in persistence that allows a structural interpretation of the approaches found in the empirical literature on aggregation and the real exchange rate. In particular, we reconcile the apparently conflicting findings that gave rise to the 'PPP strikes back' debate (Imbs et al. [2005a, b] and Chen and Engel [2005]).